Post on 19-Jul-2022
F. Peña-Benítez
Crete Center for theoretical physics
Kolymbari, sept 2014
UNDERSTANDING THE DYNAMICS OF THE CHIRAL VORTICAL EFFECT
based on: arXiv:1312.1204
K. Landsteiner, E. Megías, F. P-B
OUTLINE• the chiral magnetic effect (CME) and quantum anomalies
• time evolution
• the chiral vortical effect (CVE)
• holographic model
• time evolution
• consistency checks
• physical implications
2
MOTIVATION• quantum anomalies are interesting
• mixed gravitational anomaly could be measure for the first time
• non dissipative transport
• the non dissipative nature is not destroyed at high temperature (could be powered in some cases)
• macroscopical effect of a purely quantum property
• origin of the charge separation observed in the QGP???
• possibly realized in Weyl semi-metals
• non renormalization theorems
• …
TA
TB
TC
Aµ
Aν
Aρ
+++++
In a field theory global symmetries can be violated at the quantum level
Theories with massless fermions have axial and vector global symmetries
It is not possible to preserve in the QFT (even dimensions) both symmetries
In (3+1)d there are two types of axial anomalies
what do we know about quantum anomalies?
ANOMALIES
4
~B
electric charge flow
axial charge flow
momentum flow
~!electric charge flow
axial charge flow
momentum flow
~B
electric charge flow
axial charge flow
momentum flow
~ja =
0
@~je~j5~j"
1
Aji" = T ticharge transportchirality transportenergy transport
Kubo formulas
Kubo formulas
~ja =
0
@~je~j5~j"
1
Aji" = T ticharge transportchirality transportenergy transport
Ba = lim
qz
!0
i
qzGR
jxa
jy (0, qz)
~ja = Ba~k ~A
t,y
x
~k
z
Kubo formulas for anomaly induced transport~J = V r ~v
uµ = (1,~0) ! uµ = gtµ
Agi gti
~J = V r ~Ag
ds
2 = (µ + hµ)dxµdx
V = limqz
!0
i
qzGR
jx T ty
(0, qz)
[Amado, Landsteiner & F. P-B, JHEP.1105 (2011)]
= limk!0
1
k
~jc = Tc~P+
J iA T 0j
q
q + k
T 0i =i
4 (0@i + i@0)P+
Va =
1
162
2
4X
b,c
Tr(TaHb, Hc)µbµc +22
3T 2Tr(Ta)
3
5
Bab =
1
42
X
c
Tr(TaTb, Hc)µc
[Landsteiner, Megías & F. P-B, PRL. 107 (2011)]
[Kharzeev & Warringa, PRD.80 (2009)]
free fermions &
anomaly induce transport
µf =X
a
qfaµa , Ha = qfafg
TA
TB
TC
Aµ
Aν
Aρ
TA
hµν
hλβ
Aρ
ba Tr[Ta]cabc Tr[TaTb, Tc]
in the case of interest for QCD U(1)VxU(1)A
rµjµ = 0
rµjµ5 =
1
162Fµ F
µ +1
3842µR↵
µR
↵
Solutions for U(1)VxU(1)A
free fermions conductivities
BA,(0) =
1
22
8<
:
µ5
µµµ5
, A = e, 5,
B5
A,(0) =1
22
8<
:
µµ5µ2+µ2
52 + 2T 2
6
τ = 0.1/T
τ = 0.2/T
τ = 1/T
t/τ
j(t)σ0B0
151050-5
0.4
0.3
0.2
0.1
0
σ′′χ
σ′χ
ω/T
σχ(ω)σ0
3020100
1
0.5
0
5 10 15 20 25 30
Ω
T
"0.5
0.5
1.0
Σ
Σ0
Μ!T%0.1
free fermions magnetic
conductivity
holographic magnetic
conductivity
B(t) =1
[1 + (t/)2]3/2B0
[Kharzeev & Warringa, arXiv:0907.5007]
[Ho-Ung Yee, arXiv:0908.4189]
Solutions for U(1)VxU(1)A
[Vilenkin, Phys. Rev. D20, 1807 (1979)]
VA,(0) =
1
22
8><
>:
µµ5µ2+µ2
52 + 2T 2
6
µ5
µ2+µ2
53 + 2T 2
3
Solutions for U(1)VxU(1)A
[Vilenkin, Phys. Rev. D20, 1807 (1979)]
free fermions conductivities
V (!) = V0 (!,0 + i!(!))
VA,(0) =
1
22
8><
>:
µµ5µ2+µ2
52 + 2T 2
6
µ5
µ2+µ2
53 + 2T 2
3
[Landsteiner, Megías & P-B. 1312.1204]
holographic modelwe need to build a model with a U(1)VxU(1)A global symmetry the vector current must be conserved the axial current must be anomalous with the mixed gauge-gravitational anomaly included
S =1
16G
Zd
5x
pg
R+
12
L
2 1
4
FMNF
MN + F
(5)MNF
(5)MN
is a vector gauge field in the bulk
is an axial gauge field in the bulk
the boundary value corresponds to a background vector source
the boundary value corresponds to a background axial source
AM (r, x)
A
(5)M (r, x)
A
(5)µ (1, x)
Aµ(1, x)
is the bulk metric the boundary background metric hµ(1, x)gMN (r, x)
holographic modelwe need to build a model with a U(1)VxU(1)A global symmetry the vector current must be conserved the axial current must be anomalous with the mixed gauge-gravitational anomaly included
is a vector gauge field in the bulk
is an axial gauge field in the bulk
the boundary value corresponds to a background vector source
the boundary value corresponds to a background axial source
AM (r, x)
A
(5)M (r, x)
A
(5)µ (1, x)
Aµ(1, x)
is the bulk metric the boundary background metric hµ(1, x)gMN (r, x)
S
ano
=
Zd
5x
pg
h
MNPQR
A
(5)M
3F
(5)NP
F
(5)QR
+ F
NP
F
QR
+ R
A
BNP
R
B
AQR
i
holographic modelwe need to build a model with a U(1)VxU(1)A global symmetry the vector current must be conserved the axial current must be anomalous with the mixed gauge-gravitational anomaly included
S
ano
=
Zd
5x
pg
h
MNPQR
A
(5)M
3F
(5)NP
F
(5)QR
+ F
NP
F
QR
+ R
A
BNP
R
B
AQR
i
5(S + S
ano
+ S
bound
) /Z
@
d
4x
ph5
µ
3F
(5)µ
F
(5)
+ F
µ
F
+ R
↵
µ
R
↵
holographic modelwe need to build a model with a U(1)VxU(1)A global symmetry the vector current must be conserved the axial current must be anomalous with the mixed gauge-gravitational anomaly included
S
ano
=
Zd
5x
pg
h
MNPQR
A
(5)M
3F
(5)NP
F
(5)QR
+ F
NP
F
QR
+ R
A
BNP
R
B
AQR
i
5(S + S
ano
+ S
bound
) /Z
@
d
4x
ph5
µ
3F
(5)µ
F
(5)
+ F
µ
F
+ R
↵
µ
R
↵
rµjµ5 =
1
162Fµ F
µ +1
3842µR↵
µR
↵
++
ds
2 =r
2
L
2
f(r)dt2 + d~x
2+
L
2
r
2f(r)
dr
2
A =
µ r2H
r2
dt
A(5) =
µ5 r2H
r2
dt
Q =r2Hp3µ Q5 =
r2Hp3µ5
AdS Reissner-Nordström blackhole
Numerical vortical conductivity seems to agree with the free case
V (!) = V0 (!,0 + i!(!))
vortical conductivities
0 7 14-3. ¥ 10-6
0
6. ¥ 10-6
0 5 10 15 20 250.0
0.2
0.4
0.6
0.8
1.0
w
T
s5VHw, 0L
k
T!
2 Π
5
k
T!Π
25
k
T!Π
250
Hydro prediction
0 2 4 6 8
0.0
0.2
0.4
0.6
0.8
1.0
4ΠTΩ
k2
Σe!!Ω, 0"
Numerical vortical conductivity seems to agree with the free case
Notice the peak seems to be at
! 1
4Tk2
this coefficient coincides with the SYM diffusion
constant
V (!) = V0 (!,0 + i!(!))
vortical conductivities
HYDRODYNAMICS
Tµ = (+ P )uµu + Pgµ µ + B (Bµu +Buµ) + V
(!µu + !uµ) ,
Jµ = uµ + BBµ + V !µ
rµTµ = F µJµ
T ti = (+ P )vi + Phti ikijV (vj + htj) ikij
B Aj
J i = vi ikijV (vj + htj) ikij
BAj
ij
B
!kAj
+ (i!(+ P ) + k2)vi
i!(P + )hti
ij
V
!k(htj
+ vj
) !khxi
i!Ai
= 0
k
T!
2 Π
5
k
T!Π
25
k
T!Π
250
Hydro prediction
0 2 4 6 8
0.0
0.2
0.4
0.6
0.8
1.0
4ΠTΩ
k2
Σe!!Ω, 0"
Hydro prediction
0 2 4 6 8
0.0
0.2
0.4
0.6
0.8
1.0
4ΠTΩ
k2
Σ!
!!Ω, 0"
< T tiT tj > = ikijV
D2k4
(! + iDk2)2
< T tiJj > = < J iT tj >= ikijB
iDk2
(! + iDk2)
!M = ± 1p3Dk2 , < T tiT tj >
!M = ±Dk2 , < T tiJj >
ENERGY CONSERVATION
W =
Zd
4x
W
gµ(x)gµ(x) +
W
Aµ(x)Aµ(x)
= 0
rµ
2pg
W
gµ(x)
+rµ
1pg
W
Aµ(x)
A
(x) +1pg
W
Aµ(x)Fµ
(x) = 0
Euclidean effective action
ENERGY CONSERVATION
i(! + i)0x,0y(i!
n
= ! + i, kz
)
ikz
= 0x,zy(i!n
= ! + i, kz
)
! V (!) = 0
i(! + i)Gx,0y(i!
n
= ! + i, kz
)
ikz
kz!0
= Gx,zy(i!n
= ! + i,~k = 0)
! V (!) = 0
vanishing at zero momentum because of rotational invariance
to understand the physical implication of these result, let's play a game
0
!(s) =ik
s+ i
V (s) = V0
iDk2
s+ iDk2
J(t) = V0 k(1 eDk2t)(t)
k
TA
TB
TC
Aµ
Aν
Aρ
+++++
It is a temptation to play with this result and the characteristic numbers of the QGP
for the holographic plasma
the lifetime of the QGP is of order
J(t) = V0 k(1 eDk2t)(t) =
1
Dk2
D =1
4T
2100fm
L 10fm
T 350Mev
10fm
It is a temptation to play with this result and the characteristic numbers of the QGP
for the holographic plasma
the lifetime of the QGP is of order
However in the QGP the vorticity is not an external source and the real problem is much more complicated because the vorticity is a dynamical variable!
J(t) = V0 k(1 eDk2t)(t) =
1
Dk2
D =1
4T
2100fm
L 10fm
T 350Mev
10fm0 2 4 6 8
0.0
0.2
0.4
0.6
0.8
1.0
t
tcCVE
J!t"#!Σ
V"
k" " !t" # "k $ !t"" !t" # "k $$ !t" % $ %t%t f &'
& tcCVEΤQGP &
%%%%%
• we need to study more realistic situations because the vorticity is a dynamical quantity
• looking forward for Weyl semimetals!
• are we close to see experimental evidence of the presence of the mixed gravitational anomaly in nature?
• the existence of the anomalous conductivities is well understood already.
• now is necessary to understand the renormalization issues in presence of dynamical gauge fields
• generation of axial chemical potential mechanism in order to be able to do good predictions for QGP
BA,(0) =
1
22
8<
:
µ5
µµµ5
, A = e, 5,
B5
A,(0) =1
22
8<
:
µµ5µ2+µ2
52 + 2T 2
6
VA,(0) =
1
22
8><
>:
µµ5µ2+µ2
52 + 2T 2
6
µ5
µ2+µ2
53 + 2T 2
3
This research has been co-financed by the European Union (European Social Fund, ESF) and!Greek national funds through the Operational Program "Education and Lifelong Learning"!of the National Strategic Reference Framework (NSRF), under the grants schemes "Funding!of proposals that have received a positive evaluation in the 3rd and 4th Call of ERC Grant Schemes"!and the program "Thales"